Characterizations of analytic planar functions
type: Item
instance of: scholarly article
title: Characterizations of analytic planar functions
zbMATH Open document ID: 0836.51013
DOI: 10.1016/0097-3165(95)90008-X
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-01-25T00:00:00Z
review text: Let \(G\) and \(H\) be abelian groups. A function \(f: G\to H\) is called planar if for all \(d\in G\setminus \{0\}\) the function \(f_d: G\to H: x\mapsto f(x+ d)- f(x)\) is bijective. With every planar function one can associate an affine plane by taking as points the elements of \(G\times H\) and as lines all translates of the graph of \(f\) and all sets \(\{c\}\times H\) with \(c\in G\). Assume now that both \(G\) and \(H\) are equal to the additive group of the complex number field and that \(f\) is complex analytic. The author proves that then \(f\) is necessarily a quadratic polynomial and the resulting plane is desarguesian. An analogous result for the case that \(G\) and \(H\) are isomorphic to the additive group of a finite prime field has been proved independently by Gluck, Hiramine, and Rónyai-Szönyi.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 802363
zbMATH Keywords: planar function
zbMATH Keywords: oval
zbMATH Keywords: desarguesian plane
zbMATH Keywords: additive group of complex numbers
zbMATH Keywords: affine plane
zbMATH Keywords: quadratic polynomial
author: Burkard Polster
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 802363